Cole has two years remaining on his contract but is looking for a pay rise that will see his salary climb from around £80,000 to £130,000 a week.
Terry said: "He has been brilliant for club and country and I really want his contract to get resolved at Chelsea. I hope it gets done because it is very important we keep players like Ashley as well as adding to our squad. He has been in fantastic form.
"I will be in contact with him and the club over the summer to see if it gets done. He has fully proved himself.
"The fans love him and the players do too, we gave him our Players' Player of the Year Award.
"It's a nice boost for him as well, you can see him playing with a lot of confidence and for me is one of the best left-backs in the world. It is very important we keep him at the club."